An action \(\mu :T\times X\rightarrow X\)  of a one dimensional torus \(\mathbb {C}^\ast \)  on a d-critical scheme \(X\)  is said to leave invariant the d-critical structure \(s\)  (resp. the orientation \(K_{X,s}^{1/2}\) ) if \(\mu (\gamma )^\star (s)=s\)  (resp. \(\mu (\gamma )^{\ast }K_{X,s}^{1/2}=K_{X,s}^{1/2}\) ) for \(\gamma \in T\) . In particular, the d-critical structure of the classical truncation of a \((-1)\) -shifted symplectic scheme with a \(\mathbb {C}^\ast \) -action leaving the \((-1)\) -shifted symplectic structure invariant is \(\mathbb {C}^\ast \) -invariant. If the action is étale locally linearizable, then translating [1]} from the Zariski to the étale topology, we can then work with \(\mathbb {C}^\ast \) -equivariant critical charts. Namely, we can cover \(X\)  in the étale topology by charts \((R,U,f,i)\)  such that \(U\)  has a \(\mathbb {C}^\ast \) -action for which \(i\)  is equivariant and \(f\)  is invariant. Moreover, considering \((R,U,f,i)\)  and \((S,V,g,j)\)  two \(\mathbb {C}^\ast \) -equivariant critical charts, and \(x\in R\cap S\) , one has étale restrictions \((R^{\prime },U^{\prime },f^{\prime },i^{\prime })\rightarrow (R,U,f,i)\)  and \((S^{\prime },V^{\prime },g^{\prime },j^{\prime })\rightarrow (S,V,g,j)\)  with \(x\in R^{\prime }\cap S^{\prime }\) , and \(\mathbb {C}^\ast \) -equivariant critical chart \((T,W,h,k)\)  and \(\mathbb {C}^\ast \) -equivariant embeddings \(\Phi :(R^{\prime },U^{\prime },f^{\prime },i^{\prime })\hookrightarrow (T,W,h,k)\)  and \(\Psi (S^{\prime },V^{\prime },g^{\prime },j^{\prime })\hookrightarrow (T,W,h,k)\) .
